# The Hopf monoid and the basic invariant of directed graphs

## TL;DR

This paper introduces a Hopf monoid structure for directed graphs, connecting it to generalized permutahedra and showing that its basic invariant aligns with the strict chromatic polynomial, enriching the combinatorial framework.

## Contribution

It defines a new Hopf monoid for directed graphs and links its invariant to known chromatic polynomials, expanding the algebraic understanding of graph invariants.

## Key findings

- The Hopf monoid of directed graphs embeds in the Hopf monoid of generalized permutahedra.
- The basic invariant of this monoid matches the strict chromatic polynomial.
- Establishes a new algebraic perspective on graph invariants.

## Abstract

Aguiar and Ardila defined the Hopf monoid GP of generalized permutahedra and showed that it contains many submonoids that correspond to combinatorial objects. They also give a basic polynomial invariant of generalized permutahedra, which then specializes to the submonoids. We define the Hopf monoid of directed graphs and show that it also embeds in GP. The resulting basic invariant coincides with the strict chromatic polynomial of Awan and Bernardi.